Uniform retractions onto a probability measure algebra and an application to topological groups

Abstract

We prove that the indicator functions form a uniformly continuous retract of the set of $[0,1]$-valued functions in $L^2([0,1])$. Consequently, the Lebesgue probability measure algebra, equipped with the symmetric-difference metric, is an injective uniform space. The construction uses dyadic conditional expectations and a common exponential coupling to obtain Lipschitz maps into the indicator functions that approximate the identity uniformly on that set. As an application, we construct a Polish topological group whose left uniform space is injective but which admits a bounded left uniformly continuous real-valued function that is not right uniformly continuous. This answers the injective part of a question of Bouziad and Troallic in the negative. The group is a semidirect product of two copies of the measure algebra and is also bounded, locally finite, and of exponent at most four.